Dielectric sphere in point charge electric field

[paweld]

I'm interested if it's possible to compute potential inside and outside
dielectric sphere (with permittivity $$\varepsilon_1$$) which is
placed in field of point charge q lying outside the sphere where
permittivity is $$\varepsilon_2$$. I've tried to solve it
using Legendre polynomilas but I obtain infinite sum which
I cannot interpretate as images (maybe I made a mistake).
So this problem cannot be solve using methods of images?

 

[Meir Achuz]

Images won't work. Your LP sum is the standard solution.

 

[DrDu]

Images work, but there are infinitly many of them!

 

[paweld]

Thanks for answers. It's quite strange that such apparently simple
problem has only solution in terms of infinite sum.

 

[Meir Achuz]

sin x is also calculated by an infinite sum.

 

[paweld]

indeed, for computer it doesn't matter